CONTEXTUALIZING THE ACTOR AND THE PLAY:

A CASE STUDY OF THIRD GRADE STUDENTS DEVELOPING UNDERSTANDINGS OF PLACE VALUE

Janet Bowers

San Diego State University

Jbowers@Sunstroke.sdsu.edu

Recent trends in mathematics education research reflect the view that learning is an inherently social and cultural activity. This shift can be seen in the rise of studies conducted using social constructivist and situated perspectives (cf., Beach, 1995; Boaler, 2000; Cobb, Stephan, McClain, & Gravemeijer, in press; Greeno, 1997; Lave, 1997; Saxe, Gearhart, & Seltzer, 1999). On one hand, this focus enables researchers to situate learning in forms of social co-participation (Lave & Wenger, 1991). On the other hand, it can be interpreted as minimizing the importance of individual students’ reasoning (Anderson, Reder, Greeno, & Simon, 2000; Cobb, et al., in press). The objective of this paper is to develop an analysis that is based on the situated perspective, but also seeks to place individual actors’ progress within the context of the play.

Theoretical framework

Identifying practices using the situated perspective. One advantage of the situative perspective is that can be used to document differences between social practices that are established among groups in different settings. For example, in their seminal work, Lave & Wenger (1991) described different apprentice relationships found in four contrasting communities of practice: midwives, tailors, quartermasters, and alcoholics. Their goal was to compare different practices and participation structures among the groups, but not to document differences occurring within any one community. In situative studies focusing on schooling, Boaler (2000) and Beach (1995) each conducted studies that contrasted forms of arithmetical reasoning developed by two groups of students who participated in different learning practices in different school settings. In each of these studies, the researchers took participation in two broad forms of collective activity as the primary unit of analysis.

The approach taken in this analysis builds on the idea of identifying practices within a community, but also attempts to account for individual learning that occurs among members within a group. Following the situated perspective, learning is defined in terms of the process by which students actively reorganize their ways of participating in classroom practices. However, the major premise of this report is that the relation between individual students’ learning and the evolution of communal practices is viewed as reflexive (Cobb, 1994). Students contribute to the evolution of the classroom practices that constitute the immediate social situation of their mathematical development as they participate in the evolving practices of the culture. The methodological implication of this reflexivity is that research must document the emergence of specific mathematical practices and then identify different ways in which students participate in them.

Psychological constructs for assessing individual participation. The question of how to assess the ways in which individuals contribute to the classroom practices involves developing psychological constructs to describe students’ mathematical understandings. In the current study, the students were learning concepts of place value. The constructs that were developed to assess students’ understandings were based on the work of Cobb & Steffe (1983) who identified several ways in which students solve tasks involving collections. For the purposes of this study, the list was narrowed to three main distinctions: (1) creating numerical composites, (2) creating composite units, and (3) using part-whole reasoning (Cobb & Steffe, 1983). These may best be described by considering different ways that students solve the task of figuring out how many groups of ten crayons could be obtained from a bag containing 360 crayons. Students who are creating numerical composites and composite units often begin by counting by tens from 10 to 360 and keeping track using their fingers. Upon reaching 360, students who have created numerical composites state that the answer is 360 rather than 36. When asked to explain, they cannot reconcile how their fingers relate to countable groups of ten. In contrast, students who have created composite units are able to explain that the answer is 36 because they have counted 36 groups of ten items. Students who partition the collection into a group of 300 (containing 30 composite units of ten) and 60 (containing 6 composite units of ten) are reasoning in terms of part-whole relations.

Setting, Data, and Methodology

Setting and data collection. The setting for the teaching experiment was a third-grade classroom in a public suburban school in the southern United States. The class consisted of 23 students (14 boys, 9 girls). Seven of the students (30%) were members of ethic minority groups (4 African American boys, 2 African American girls, and 1 Indian girl). The data corpus for the entire study consists of: videotapes from each lesson, field notes, copies of all students’ written work, and protocol sheets from pre- and post- individual interviews conducted with all of the students just before and after the teaching experiment.

Instructional Sequence. The goal of the Candy Factory sequence (developed by Cobb, Yackel, & Wood, 1992) was to support students’ construction of increasingly efficient but not necessarily standard algorithms that reflected their developing understanding of place value numeration. To this end, the students first pretended to be workers in a candy factory where ten candies were packed in a roll, and ten rolls in a box. In subsequent instructional activities, the students developed a variety of ways of symbolizing the process of combining and separating quantities of boxes, rolls, and pieces. These included drawing pictures, making tally marks, and writing numerals.

Methodology. The method used to collect and analyze the data involved two steps. First, researchers documented the overall mathematical practices that emerged in the classroom over the course of the 9-week teaching experiment (See Bowers, Cobb, & McClain, 1999 for a full description of the five mathematical practices). The critical issue in identifying a mathematical practice was that it was not defined by conducting a modal trend analysis to see when a majority of students began to change their ways of acting. Instead, practices were distinguished by seeing how contributions made during whole-class discussions served as catalysts for shifting the types of explanations and justifications that eventually became taken-as-shared (cf., Cobb, Yackel, & Wood, 1992).

The second step of the analysis method involved using the psychological constructs described above to analyze how individual students were thinking about place value as they participated in the identified practices. To this end, four target students were chosen by the research team (on suggestion from the teacher) at the outset of the teaching experiment. These four students, who together to represented a variety of different skill levels, were interviewed and carefully observed by the research team over the course of the teaching experiment.

Results

The mathematical practices

Table 1 includes a brief description of each identified practice, the accompanying ways of justifying that were established during each practice, and the precipitating event that initiated the practice (i.e., the particularly significant observation or suggestion made by a student that contributed to the shift in ways that students explained and justified their reasoning).

Table 1. Mathematical practices that emerged over the course of the teaching experiment.

Practice	Methods of Explanation and Justification	Precipitating Event
I. Enumerating a given collection	Students enumerated pre-drawn collections of boxes, rolls, and pieces. Justifications involved counting or iterating units.	(first practice, no precipitating event)
II. Creating different arrangements	Students drew different ways that a given number of candies could be packed. Justification involved counting by hundreds, tens, and ones.	Students described the conjecture that collections of candies that were “all packed up” (in canonical form) were easiest to enumerate.
III. Transforming quantities	Students continued to draw different ways, but changed their ways of justifying from packing to interpreting the results of transformations preserve quantity.	Students noted that it was possible to anticipate new collections without actually drawing them and that the number of pieces would have to be the same.
IV. Adding and subtracting	Students drew pictures to show addition and subtraction activities. Ways of justifying involved counting their drawings, and then recording their answers on an inventory form.	Students used their insights about imagining unpacking to facilitate the process of sending out orders of candy.
V. Partitioning and recombining collections	Students were again enacting addition and subtraction tasks, but anticipated unpacking rather than making drawings.	Students established different (but not equally efficient) ways of notating transformations.

Individual ways of participating in the mathematical practices

The constructs for gauging students’ participation in classroom practices can be arranged in order of increasing levels of sophistication from creating numerical composites to composite units to reasoning in terms of part-whole relations. Based on this ordering, the chart in Figure 1 indicates that, although they shifted at different times and to differing degrees, each of the four target students did develop increasingly sophisticated ways of thinking about place value through the course of the teaching experiment.

Martine participated in mathematical practices I and II by interpreting the rolls as numerical composites. For example, during one whole-class discussion in which students were asked to draw different ways that 532 pieces could be partially packed, Martine drew 50 rolls on the board but described them to the class as “500 rolls.” Although his comment may have meant 500 pieces arranged in rolls (as Carolyn, another target student suggested), this response, along with other written work, indicates his construction of numerical composites rather than composite units. As he participated in practice III, he began to interpret the rolls and boxes in terms of composite units. This shift was initiated when he began to reorganize his activity of creating different arrangements of candies by unpacking previous ones rather than starting from scratch each time.

Wilma also made a shift from interpreting the rolls and boxes as numerical composites to composite units as she participated in the third math practice. For example, when drawing different ways to show how a collection could be partially packaged, she initially justified each of her new solutions by counting each item (by hundreds, tens, or ones accordingly) whenever she unpacked a new box or roll. As she participated in practice III, she curtailed her counting when she realized that she did not have to recount because the total quantity of candies did not change.

Carolyn interpreted the boxes and rolls in terms of composite units early on as she participated in practice I. As she participated in the activity of drawing combinations of 532 candies described earlier, her drawings on the blackboard and accompanying justifications indicated that she was reasoning in terms of part-whole relations. That is, unlike Martine and Wilma, Carolyn participated in class discussions by explaining that she imagined a series of unpacked boxes before drawing any of them. This indicates that she interpreted collections as both one whole quantity and as composite units that could be partitioned into constituent parts simultaneously.

Bob was a vocal participant who was actively involved in negotiating two of the shifts in the mathematical practices that occurred over the course of the teaching experiment. As Figure 2 indicates, Bob participated in practices I and II by interpreting the boxes, rolls, and pieces in terms of numerical composites. The precipitating event that contributed to his construction of composite units was the observation he made in class that he could anticipate the results of multiple packings without actually carrying out the transformations. This observation, and a similar one regarding addition and subtraction, led the class to agree on a notation system to keep track of the results of each imagined transformation and served as a catalyst for negotiating the emergence of new practices.

Discussion and Conclusions

The overall objective of this analysis has been to document how individual students contributed to, and were consequently affected by, the emergence of mathematical practices. This approach builds on a situative view of learning as participation in social practices, but also attempts to account for how shifts in the mathematical practices influenced and were influenced by individual students’ participation. One advantage to this approach is that it provides teachers and curriculum developers with a clearer picture of what activities and discussions initiated shifts in the ways that students engaged in various tasks. The analyses of the practices indicated that two major transitions emerged. The first occurred during the transition from practices II to III, when the students reorganized their activity from drawing pictures to create different configurations to realizing that packing and unpacking a collection did not change the total quantity of candies. This is not to say that all students suddenly shifted their ways of acting at the same time or in the same way. Instead, the precipitating events involved noticing that transformations preserved quantity and that a collection that was “all packed up” could be most easily counted. Taken together, these conversations supported students’ efforts to make their work more efficient. This can be seen, for example, in the target students’ efforts to shift from justifying their answers by circling ten rolls or pieces and describing packing to implicitly agreeing that each new arrangement was the result of transforming a previous arrangement.

In closing, the underlying assumption of this analysis was based on the view that learning is a social process in which students reorganize the ways that they participate in activities. This report has revealed that the two critical shifts in mathematical practices were each precipitated by conversations that involved collective anticipation such that students began describing what might happen rather than what did happen and using sophisticated justifications that became taken-as-shared. These observations provide researchers with insights regarding the types of arguments that raise the level at which students justify their answers, and also provide a methodology for placing each actor’s contributions within the overall play.

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